Sycamore family: Difference between revisions

The '''sycamore family''' of [[regular temperament|temperaments]] tempers out the [[sycamore comma]] ({{monzo|legend=1| -16 -6 11 }}, [[ratio]]: 48828125/47775744).  

The head of this family is [[5-limit]] sycamore. Its [[generator]] is a [[25/24|classic chromatic semitone]], and stacking six of these gives 5/4 (and hence five 6/5) and eleven give 3/2. [[94edo]] [[support]]s sycamore, and 5\94 is recommendable as a generator. It can be described as the 19 & 94 temperament, and uses a decidedly flat version of the chromatic semitone as a generator. [[Mos]] of 18 or 19 notes to the octave give enough room for sycamore's triads, but 37 notes can be tried by the adventurous.

 

[[Subgroup]]: 2.3.5

{{Mapping|legend=1| 1 1 2 | 0 11 6 }}

[[Optimal tuning]]s:  

* [[WE]]: ~2 = 1200.6031{{c}}, ~25/24 = 63.8108{{c}}

{{Optimal ET sequence|legend=1| 18, 19, 56, 75, 94, 207c, 301c }}

[[Badness]] (Sintel): 4.93

The second element of the [[normal forms #Normal forms for commas|normal comma list]] for septimal sycamore is [[875/864]], the keema, and it also tempers out [[686/675]], the senga, and [[3136/3125]], hemimean. It may also be called the 19 & 56 temperament. This may also be used as the name for the temperament obtained by adding [[100/99]] to sycamore's commas, giving undecimal sycamore, where 10 generator steps reaches 16/11, 11 reach 3/2, and 15 give 7/4, adding a considerable dose of 11-limit harmonies to the 19-note mos. [[75edo]] is an excellent tuning for 7-limit sycamore, and [[56edo]] for the 11-limit version.

[[Subgroup]]: 2.3.5.7

{{Mapping|legend=1| 1 1 2 2 | 0 11 6 15 }}

* [[WE]]: ~2 = 1200.7208{{c}}, ~25/24 = 64.0334{{c}}

: [[error map]]: {{val| +0.721 +3.133 -0.672 -6.884 }}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~25/24 = 64.0496{{c}}

: error map: {{val| 0.000 +2.591 -2.016 -8.082 }}

{{Optimal ET sequence|legend=1| 18, 19, 56, 75d }}

[[Badness]] (Sintel): 1.57

Mapping: {{mapping| 1 1 2 2 4 | 0 11 6 15 -10 }}

Optimal tunings:  

* CWE: ~2 = 1200.0000{{c}}, ~25/24 = 64.2505{{c}}

{{Optimal ET sequence|legend=0| 18, 19, 37, 56 }}

Badness (Sintel): 1.85

Mapping: {{mapping| 1 1 2 2 4 3 | 0 11 6 15 -10 13 }}

Optimal tunings:  

* CWE: ~2 = 1200.0000{{c}}, ~25/24 = 64.2853{{c}}

{{Optimal ET sequence|legend=0| 18, 19, 37, 56 }}

Badness (Sintel): 1.42

Septimal sycamore sharpens the fifth from where it stands in the 5-limit, and lowers accuracy in order to reach 7-limit harmonies. If we retain tunings approximately (e.g. 94edo) or exactly those of Carlos Beta, we get the 19 & 94 temperament, betic, for the 7-limit. This adds [[225/224]] to the sycamore comma. The Carlos Beta tuning, with pure fifths, is a good tuning choice, but 94 or 113 equal are as well. Betic extends to the 11-limit upon addition of [[385/384]] or [[540/539]] to the list of commas, which means it supports both 7 and 11-limit marvel.  

[[Subgroup]]: 2.3.5.7

{{Mapping|legend=1| 1 1 2 1 | 0 11 6 34 }}

[[Optimal tuning]]s:  

* [[WE]]: ~2 = 1200.6891{{c}}, ~25/24 = 63.7773{{c}}

{{Optimal ET sequence|legend=1| 19, 56d, 75, 94, 113, 320cc, 433ccd }}

[[Badness]] (Sintel): 1.77

Mapping: {{mapping| 1 1 2 1 5 | 0 11 6 34 -29 }}

Optimal tunings:  

* CWE: ~2 = 1200.0000{{c}}, ~25/24 = 63.7796{{c}}

{{Optimal ET sequence|legend=0| 19, 75, 94, 207c }}

Badness (Sintel): 1.88

Mapping: {{mapping| 1 1 2 1 5 2 | 0 11 6 34 -29 32 }}

Optimal tunings:  

* CWE: ~2 = 1200.0000{{c}}, ~25/24 = 63.7702{{c}}

{{Optimal ET sequence|legend=0| 19, 75, 94, 113, 207c }}

Badness (Sintel): 1.34

[[Category:Catalogs of rank-2 temperaments]]